The Many-Electron System in the Forward, Exchange and BCS Approximation

The nonrelativistic many-electron system in the forward, exchange and BCS approximation is considered. In this approximation, which is still quartic in the annihilation and creation operators, the model is explicitly solvable for arbitrary space dimension d. The partition function and the correlation functions are given by finite-dimensional integral representations. Renormalization effects as well as symmetry breaking can be seen explicitly. It is shown that the usual mean field approach, based on approximating the Hamiltonian by a quadratic expression, may be misleading if the electron-electron interaction contains higher angular momentum terms and the space dimension is d=3. The perturbation theory of the solvable model is discussed. There are cases where the logarithm of the partition function has positive radius of convergence but the sum of all connected diagrams has radius of convergence zero implying that the linked cluster theorem is not applicable in these cases.